The Barking Dog Test For Divergence

Several people in my calculus class continue to have difficulty understanding the implications of the Nth Term Test, which states that if the limit as n approaches infinity of the nth term of the series does not = 0, the series diverges. Again and again, people interpret this to mean that if the limit is 0, that the series converges. (They do seem to understand why having a limit of 0 is a necessary condition for the series to converge.) However, a limit that = 0 tells you nothing about the convergence or divergence of the series.

Fundamentally, people are confusedly thinking that to fail to disprove something is the same as to confirm it. This is a not uncommon logic problem.

During class, I thought of an analogy to the Nth Term Test that I will call the Barking Dog Test.

A storage container/enclosed box has either a cat or dog in it, and it is your job to identify whether it is a cat or dog, and if it is a cat, what specific kind (so that, say, the boxes of various types of cats can be placed in the correct location in your storage facility and the boxes of dogs can be sent to another facility that deals in dogs).

There are many different, sometimes complex ways to determine if the animal is a cat, and what kind of cat, but one easy way to see if the animal is definitely a dog, not a cat. We will use the fact that no cat will ever respond with a bark when offered the opportunity to go on a walk; a cat will always remain silent (Bark = 0).

If you say to the animal in the box, “Do you want to go on a walk? Huh? Huh?” and the animal responds with a bark (Bark does not = 0), then the animal is a dog, not any kind of cat, and no further testing needs to be done.

However, if the animal responds with silence (Bark = 0), it is still unknown whether it is a cat or a dog. It could be some sort of cat but could also be a dog who is deaf, mute, asleep, or just uninterested in walks. To distinguish between these possibilities, you will need to apply your other methods. All the Barking Dog Test tells you is that if the animal barks (Bark not = 0), it is definitely a dog.

Similarly, if the limit of the nth term does not = 0, the series is definitely divergent. Otherwise, you need to check for convergence using other tests.

(Please note, the Barking Dog Test is not to be confused with either the Curious Incident of the Dog in the Night-time Test or the Shroedinger’s Cat Test, neither of which are applicable to the cat/dog sorting job you have been hired to do. You can muck around with mystery novels and quantum physics on your own damn time.)